Ein starker Normalisationssatz für die bar-rekursiven Funktionale

A unifying framework for continuity and complexity in higher types

“…Remark 2.2. The above formulation of bar recursion as a rewrite system is inspired by Berger [5], which in turn uses a trick due to Vogel [37]. There bar recursion is considered in its general form, where the type of a and the output can be arbitrary.…”

Strong normalisation for applied lambda calculi

“…More precisely, since turning the equation above into a rewrite rule would clearly not be strongly normalising, we will have to work with the following minor variation of modified bar recursion. We replace the conditional expression on the right hand side by a call of an auxiliary constant Ψ with an extra (boolean) argument in order to force evaluation of the test k < |s| before the subterm Φygh(s * x) may be further reduced (Vogel's trick [Vog85]). …”

“…The difficulty in proving normalisation for bar recursion and similar recursion schemes lies in the fact that these schemes do not use a recursive descent along some kind of wellfounded structural ordering, but rather rely on continuity arguments and the ability to construct, in a suitable model, infinite sequences by nonconstructive choices. Since the computability method amounts to the construction of a syntactic model (built from strongly normalising terms) which does not satisfy these requirements, one needs to enrich the model, either by introducing infinite terms [Tai71,Vog85], or by building the model from sets of terms instead of single terms [Bez85]. These modifications, which work for Spector's bar recursion, seem to fail, however, for other recursion principles which also rely on continuity and choice and which occur in recent work on computational interpretations of classical choice and related principles [BBC98,BO05,Ber04a].…”

Continuous semantics for strong normalisation

“…The problem of proving strong normalisation for typed λ-calculi and higher type rewrite systems has been studied extensively in the literature: see, for example, Tait (1971), Girard (1972), Troelstra (1973), Luckhardt (1973), Vogel (1976), Coquand (1985), Bezem (1985), Geuvers and Nederhof (1991), Barendregt (1992), van de Pol and Schwichtenberg (1995), McAllester et al (1995), Barbanera et al (1997), Blanqui et al (1999) and Matthes (2001).…”

“…Remark. In Tait (1971), Vogel (1976), Luckhardt (1973) and Bezem (1985), strong normalisation is proved for BR formulated in a combinatorial calculus. Our result is slightly stronger since we work in a λ-calculus framework that allows more reductions.…”